Random coefficient models best fit data which are nested/clustered
The correlation within the group can be analyzed
Unlike indepencence assumption in basic regression models
Also disperities between the groups can be quantified
AKA:
Hierarchical models
Multilevel models
Mixed-effect models
Random-effect models
Variance-component models
Partial pulled models
In the absence of a Bayesian hierarchical model, there are two approaches for this problem:
Independently compute influence of factors on the responce variable for each group (no pooling)
Compute an overall average, under the assumption that every group has the same underlying average (complete pooling)
Complete pooling and No pooling
Data observations are independent
Underlying characteristics is combined to a grand mean
One homogeneous model is fitted
Resulting model also known as Partial pooled model
Partial pooled model/Random coefficient
Data observations are nested to their group
Underlying characteristics are nested in groups
Heterogeneous models are fitted
Modeling the dynamic characteristics of traffic conditions
Investigate disperity associated with lane lateral loaction
Also, the effect of day of the week
Dynamic Modeling of traffic conditions
Only current traffic regime has influence on the next regime
Mathematically:
\[P_{ij}=(P_{t+1}=j|P_{t}=i)\]
\[P_{ij} = \left[\begin{array} {rr} P_{ff} & P_{fc} \\ P_{cf} & P_{cc} \\ \end{array}\right] \]
\[\sum_{j=1}^{2} P_{ij}=1\]
\[P_{ij} = \left[\begin{array} {rr} 1-P_{fc} & P_{fc} \\ 1-P_{cc} & P_{cc} \\ \end{array}\right] \]
Time-varying effect
Regression to model Markov chain Transition probability
Model is more flexible
Heterogeneous issue by time is addressed
Lane and day of the week varying-effect
Random coefficient models
Address heterogeneous characteristics associated by lane lateral location
Address heterogeneous characteristics associated by day of the week
\[ Y_{ij} = Bernoulli(\pi_{ij})\] \[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \beta X + \epsilon_{k}\]
\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]
\[ \sigma_{1} \sim unif(0, 100)\] \[ \beta \sim N(\mu=0, \sigma = 100)\] \[ \epsilon_{k} \sim N(\mu=0, \sigma = \sigma_{k})\] \[ \sigma_{k} \sim halfcauchy(0, 5)\]
\[ Y_{ij} = Bernoulli(\pi_{ij})\]
\[ \pi_{ij} = logit^{-1}(\eta_{ij})\] \[ \eta_{ij} = \alpha_{j} + \epsilon_{k}\]
\[ \alpha_{j} \sim N(\mu_{1}, \sigma_{1})\] \[ \mu_{1} \sim N(\mu=0, \sigma = 100)\]
\[ WAIC = -2*lppd + 2*pWAIC\]
where,
lppd is the log point-wise posterior density
pWAIC is the effective number of parameters
| Random coefficient | Lane lateral location |
|---|---|
| Median lane | 46.72 |
| Inner-left lane | 51.42 |
| Inner-right lane | 50.93 |
| Shoulder lane | 57.26 |
| Overall | 52.23 |
No significant difference in model fit between CIRS and RC for the SCR data
RIS was the best in fitting Breakdown data
- The results show that there are considerable evidence that there are heterogeneity characteristics of evolution characteristics associated with different lane lateral location.